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Curl of a Vector Field
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Divergence of a Vector Field
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Gradient of a Scalar Field
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Properties of Transposes
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The Transpose of a Matrix
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Cartesian Coordinate
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Transform from Cartesian to Cylindrical Coordinate
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Divergence Theorem/Gauss' Theorem
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Stokes' Theorem
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Definition of a Matrix
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Math Formulas
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Linear Algebra
> Properties of Transposes
Properties of Transposes
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If
A = a
_{ij}

be a matrix of order m × n, then the matrix obtained by interchanging the rows and columns of
A
is known as the transpose of
A
. It is represented by
A
^{T}
.
Hence if
A = a
_{ij}

of order
m × n, then
A
^{T}
= a
_{ij}

of order
n × m.
Example:
If
, then
The following properties are valid for the transpose:
· The transpose of the transpose of a matrix is the matrix itself:
(A
^{T}
)
^{T}
= A
·
Transpose of a scalar multiple
:
The transpose of a matrix times a scalar (
k
) is equal to the constant times the transpose of the matrix:
(kA)
^{T}
= kA
^{T}
·
Transpose of a sum:
The transpose of the sum of two matrices is equivalent to the sum of their transposes:
(A + B)
^{T}
= A
^{T}
+ B
^{T}
·
Transpose of a product:
The transpose of the product of two matrices is equivalent to the product of their transposes in reversed order:
(AB)
^{T}
= B
^{T}
A
^{T}
· The same is true for the product of multiple matrices:
(ABC)
^{T}
= C
^{T}
B
^{T}
A
^{T}
.
Example 1:
Find the transpose of the matrix
and verify that
(A
^{T}
)
^{T}
= A
.
Solution:
The transpose of matrix A is determined as shown below:
And the transpose of the transpose matrix is:
Hence
(A
^{T}
)
^{T}
= A.
Example 2:
If
and
, verify that
(A ± B)
^{T}
= A
^{T}
± B
^{T}
.
Solution:
and the transpose of the sum is:
The transpose matrices for
A
and
B
are given as below:
And the sum of the transpose matrices is:
Hence
(A ± B)
^{T}
= A
^{T}
± B
^{T}
.
Example 3:
If
and
, verify that
(AB)
^{T}
= B
^{T}
A
^{T}
.
Solution:
The product of A and B is:
And the transpose of (AB) is:
If we take the transpose of A and B separately and multiply A with B, then we have:
Hence
(AB)
^{T}
= B
^{T}
A
^{T}
.
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